Cosmology

Gauss believed in the afterlife because he considered it wasteful for there not to be one. People come up with every kind of rationalization. For that belief you would have to believe in a God to begin with, because there is no injunction against nature for being wasteful. Indifferent nature has no motive. One could say God was merely a personification of indifferent nature by humans in an attempt to give it some human qualities of mercy and justice for those we love, and the power to destroy those we do not love. The destruction of one's enemies has been an important role for God throughout history. It made him a star.

I like Christmas. I even love Christmas. Jesus has a lot more to recommend him than Mohammed. Our modern vision of Christmas was created by Johnny Marks and Montgomery Wards, but still I like it. It is a beautiful fairytale. The core of it could even be true.

From one point of view modern Christmas is an ugly capitalist scheme to get people into stores. A pretty package for greed. On the other hand it is a beautiful tradition filled with merriment and good cheer.

I feel sorry for kids today that they cannot experience Christmas the way kids fifty years ago did. Most of the charm is now gone from the tradition. I see kids, I am with them, I know it is not the same for them now. Parents and grandparents try to keep the tradition alive as they knew it, but it is a losing battle against multi-cultural political correctness instituted by an army of diverse activists whose ridiculous college educations prepared them for nothing else. They majored in baloney like "public policy" and "gender studies," then they were loosed upon society. Since they now know nothing useful, they become activists, they start a board, a foundation, an institute. The world, especially our country, is so filled with useless activists I am confident I could find someone advocating for people with two rectums, if I tried hard enough. The toilets we use show how little we care after all.

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The rest of this post was so controversial I decided not to publish it during Christmas season. There is still enough to gnaw on.

It occurred to me today that the idea of our being simulations has some similarities to my own idealist viewpoint. Consciousness is behind both of them.

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Originally Posted by YesNo

It occurred to me today that the idea of our being simulations has some similarities to my own idealist viewpoint. Consciousness is behind both of them.

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If we are simulations, what the heck are we simulations of? Maybe we are simulations of beings with free will. If we are simulations, why were we only given sensitivity to small bands of light and sound? Our makers considered that enough-- but for what? And just where are these makers carting our dead off to? No one makes something they do not use in some way, even if it is only art to view. We could be the makers' art form.

Be assured of one thing, God is not going to answer any questions. We will get the answers for ourselves, or not at all. That is what we are up to and have been up to.

Why all the injunctions against "earthly" knowledge, though? What did God have against us wising up? Those parts of the Bible sound very humanly inspired to me, as in keeping your subjects kowtowing to the emperor, sultan, king.

I have to say that I believe 0% of the Bible and Koran were divinely inspired. Whatever God did was done with the creation of a universe with infinitely unfolding emergent properties. Want hints about God?--study the universe. Mathematicans and physicists--all sciences--try to study its architecture.

I am reminded of a glass being half full or half empty. It is a matter of perspective. When you say that the Bible or the Koran are 0% divinely inspired in contrast I view all texts as being divinely inspired. That would include the Bible and the Koran as well as our posts on this thread.

Simulations with free will are even closer to my view of what we are than deterministic simulations. Underlying the simulation idea is some consciousness creating and maintaining the simulation.

When you talk about "earthly knowledge", what are you referring to? Is there some Bible or Koran verse you are referring to? I have read only about 10% of these texts. If that. I am not familiar with them.

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Originally Posted by YesNo

I am reminded of a glass being half full or half empty. It is a matter of perspective. When you say that the Bible or the Koran are 0% divinely inspired in contrast I view all texts as being divinely inspired. That would include the Bible and the Koran as well as our posts on this thread.

Simulations with free will are even closer to my view of what we are than deterministic simulations. Underlying the simulation idea is some consciousness creating and maintaining the simulation.

When you talk about "earthly knowledge", what are you referring to? Is there some Bible or Koran verse you are referring to? I have read only about 10% of these texts. If that. I am not familiar with them.

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By your definition Mein Kamph was divinely inspired.

1 Corinthians 3:19

For the wisdom of this world is folly with God. For it is written, "He catches the wise in their craftiness."

1 Corinthians 19-20

For it is written, "I will destroy the wisdom of the wise, and the discernment of the discerning I will thwart." Where is the one who is wise? Where is the scribe? Where is the debater of this age? Has not God made foolish the wisdom of the world?

James 3:15

This is not the wisdom that comes down from above, but is earthly, unspiritual, demonic.

Colossians 3:2

Set minds on things that are above, not on things that are on earth.

Isaiah 44;25

Who frustrates the signs of liars and makes fools of diviners, who turns wise men back and makes their knowledge foolish?


Over and over in the Bible, thousands of times, the message is the same, one of pure control: Seek no wisdom but that found in God. Any earthly knowlege is evil and demonic.

Remember, the mighty Koran is only the size of a first book of poetry published by a typical independent press. There are not as many injunctions, but they are there. You will find them.

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Originally Posted by desiresjab

By your definition Mein Kamph was divinely inspired.

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As well as the anti-Mein Kampf texts. Someone conscious wrote them and got inspiration from somewhere. Being inspired doesn't mean the texts are infallible.

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Originally Posted by desiresjab

1 Corinthians 3:19

For the wisdom of this world is folly with God. For it is written, "He catches the wise in their craftiness."

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That sort of makes sense actually. A lot of people who think they are wise are not.

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Originally Posted by desiresjab

1 Corinthians 19-20

For it is written, "I will destroy the wisdom of the wise, and the discernment of the discerning I will thwart." Where is the one who is wise? Where is the scribe? Where is the debater of this age? Has not God made foolish the wisdom of the world?

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That one also makes sense. Do we really think we know what is going on?

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Originally Posted by desiresjab

James 3:15

This is not the wisdom that comes down from above, but is earthly, unspiritual, demonic.

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I assume this is referring to superior wisdom. Some of us are brighter than others. God should be brighter than all the rest. It makes sense to focus on what is best.

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Originally Posted by desiresjab

Colossians 3:2

Set minds on things that are above, not on things that are on earth.

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Superior wisdom again.

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Originally Posted by desiresjab

Isaiah 44;25

Who frustrates the signs of liars and makes fools of diviners, who turns wise men back and makes their knowledge foolish?

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I could come up with a similar justification for that.

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Originally Posted by desiresjab

Over and over in the Bible, thousands of times, the message is the same, one of pure control: Seek no wisdom but that found in God. Any earthly knowlege is evil and demonic.

Remember, the mighty Koran is only the size of a first book of poetry published by a typical independent press. There are not as many injunctions, but they are there. You will find them.

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I think the point of these passages is to avoid delusion and focus on what is true.

The language in all so-called wisdom literature is general enough to take in a wide sweep. I could find wise passages in Nostradamus or just about anyone else.

The point of controlling knowledge and the very definition of what knowledge is, is to control society, and the passages I quoted are examples of religion in action doing just that. People still use such Biblical injunctions to perpetuate all sorts of nonsensical beliefs not fit for modern minds.

You may interpret these texts with a liberal modern hand and think how beautiful they are, just as believers do, picking and choosing what you like and sweeping under the rug what you do not care for, but the intent of the authors had nothing to do with symbolism. You pay loyalty only to God through the temple, that was the big message. In the meantime jews are omnipresent in western academia, so they are not taking it too seriously. They are gathering up earthly knowledge. A hoard heaped by the roadside... [Joyce].

Generally it is hard to find someone who is completely wrong. There is wisdom all over the place. Some is just harder to find.

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Originally Posted by YesNo

Generally it is hard to find someone who is completely wrong. There is wisdom all over the place. Some is just harder to find.

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I am peering at every detail of Eisenstein's proof of QR, and making headway. Every detail must be accounted for and understood. More pieces are falling into place every time I focus.

My bet has changed. I now beleive I will see to the bottom of the well on QR from the Eisenstein perspective. After that, I would like to add some other perspectives, such as combinatorial and group theory approaches. But Eisenstein is not fully transparent yet.

Tiny snags.

You should be able to see to the bottom of the well.

I have been using Google Sheets to gather data about primitive roots as well as checking some papers online to get a feel for the problem. Since I need to show that 2 is a primitive root for infinitely many primes, I will have to find some way to use the information about a finite number of primes (assume all of them) for which 2 is a primitive root and reason from there that there must be another one.

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Originally Posted by YesNo

You should be able to see to the bottom of the well.

I have been using Google Sheets to gather data about primitive roots as well as checking some papers online to get a feel for the problem. Since I need to show that 2 is a primitive root for infinitely many primes, I will have to find some way to use the information about a finite number of primes (assume all of them) for which 2 is a primitive root and reason from there that there must be another one.

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I see what you want--something that proceeds along similar lines to Euclid's proof of the infinitude of primes.

Is this a proven proposition--that 2 is a primitive root for infinitely many primes--or someone's unproven conjecture?

Was this something to do with Artin's conjecture? How quickly I forget except what is in my tunnel vision.

Okay, now I have refreshed. That type of simple appearing yet intractable problem is typical of number theory. One sometimes is astonished that certain propositions which are so simple go unproven for so long. No doubt, many a doctoral dissertation has beat its head against your particular problem, which is indeed Artin's conjecture.

Good luck to both of you in your search.

Thanks, Dreamwoven!

Yes, it is Artin's conjecture, or part of it. A lot of number theory books are available as pdfs from the internet. There are more than I have time to read. Luckily for me I don't have to read all of them since they repeat themselves, just understand a few.

Eisenstein has done something extraordinary. His proof actually has nothing to do with QR, other than a method for building the right exponent to go on -1, though it does involve the primes p and q. It looks at the ratios of p and q when the prime in the numerator is multiplied by successive even numbers under the modulus of the other, with a chop function appended. In other words a function that always rounds down instead of moving to the nearest value. It would have been sufficient to find the correct parity under any circumstance, but Eisenstein is more exact than that, producing the exact exponent on -1 as the number of lattice points in the prescibed regions of his p by q rectangle. Only the (p-1) by (q-1) part interests him, containing the interior lattice points of the larger rectangle, and then only those with even coordinates.

This proof is wonderfully clever. The downside is that it will not reveal any deeper properties of numbers that help elucidate why QR works. Deeper investigations might uncover why it works. It proves what it proves--that Eisenstein's method will always find the right exponent for -1 in the Legendre symbol.

Is the computation done in polynomial time in terms of the number of digits of p and q or something slower?

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Originally Posted by YesNo

Is the computation done in polynomial time in terms of the number of digits of p and q or something slower?

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Uhhhh....I am not sure I understand the question.

Finding a quadratic residue of a given number no matter how large, should be a P time exercise. You may be referring to something like RSA encryption or Diffe-Hellman key exchange. Various encryption systems are based on particular aspects of number theory laws. It can be quadratic residues, primitive roots or mod inverses they use to "conceal" the message.

In reverse the problem gets nasty real fast, as in NP nasty. It is easy to give some quadratic residues of a number q. But given a quadratic residue, it is impossible to find q in ploynomial time, if the numbers involved are long enough.

Currently, it takes numbers of about four thousand digits length to get your bank account information encrypted securely. If some country or individual had the power to quantum compute, it has already been mathematically proven they could break our present codes in seconds.

That answered the question. I was wondering if there were a computation problem still unsolved. I suppose a quantum computer runs faster because of a potential parallel processing involved. Wouldn't a network of computers working in parallel be able to simulate such a computer?

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Originally Posted by YesNo

That answered the question. I was wondering if there were a computation problem still unsolved. I suppose a quantum computer runs faster because of a potential parallel processing involved. Wouldn't a network of computers working in parallel be able to simulate such a computer?

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That is the same question I have had. I have a hunch the answer would be "yes," provided that we link enough silicon computers together to fill the solar system, or some other great volume of space.

I went to town today and forgot graphing paper. If I graph out some p's and q's Eisenstein style, another key to QR may pop out visibly or algebraically.

I have some other studies I am stalling right now because I cannot let go of QR when I am so close. Like one of those movie bounty hunters who has pursued a particular fugitive for a long time, I cannot go for coffee now that the fugitive has been sighted. However, at my age I find I need two days rest after pursuing the most intense thinking for one day. Total forced mental focus was something I learned in math and then adapted for dummies from reading about Newton and Tesla. Now, in math I have to force it, because it hurts me to concentrate that hard on x, y and z in a foreign medium, whereas in a field like fiction writing I can stay immersed indefintely without forcing myself to, indeed without ever becoming conscious of the need to force myself to do anything.

The creative process is so much more enjoyable while it is happening, but when titantic struggles with x, y and z are over and have resulted in a surrender without terms, I find my picture of the universe is altered, and my picture of myself, as well. In this medium of math I am no natural, but I am a zealous convert.

I am lost in this discussion, never was any good with maths. This discussion of time-warps and the past and future of Space-Time may be of interest: http://www.space.com/31495-space-tim...ekly_2016-1-04

I finally reeled Eisenstein in all the way. Of course that does not mean I understand Eisenstein as well as Eisenstein did. I feel like champagne anyway.

Interesting survey article on spacetime, Dreamwoven. In particular this quote:

"It might be that space-time at very short distances takes yet another form and perhaps is not continuous," Amendola said.

Congratulations, desiresjab! I was reading this about quadratic reciprocity. It is a very elementary summary of it: http://sites.millersville.edu/bikena...-residues.html

The part I was interested in was if a prime is a primitive root for another prime it would have to be a quadratic nonresidue. I haven't tried to understand Eisenstein's proof.

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Originally Posted by YesNo

Interesting survey article on spacetime, Dreamwoven. In particular this quote:

"It might be that space-time at very short distances takes yet another form and perhaps is not continuous," Amendola said.

Congratulations, desiresjab! I was reading this about quadratic reciprocity. It is a very elementary summary of it: http://sites.millersville.edu/bikena...-residues.html

The part I was interested in was if a prime is a primitive root for another prime it would have to be a quadratic nonresidue. I haven't tried to understand Eisenstein's proof.

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You are welcome. As usual, one wonders why he didn't see it sooner. I guess it is the leap from the quadratic to the linear. -1 is a dummy base, only good for determining if the exponentiated value is positive or negative. His whole lattice graph is linear, yet it explains a quadratic law. People like Eisenstein are not really human, they just share some genes with the rest of us.

I am going to toy around with making a new encryption system. If nothing else, I will learn the present systems better. I have to come up with an appropriate function.

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There is no guarantee that quantum theory and Einsteinian physics are uniteable. Parts of Einstein are already dated, Ed Mitchell has said. What if never the twain shall meet?

If those quantum computers ever happen beyond a few qubits we may need a new encryption system.

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Originally Posted by YesNo

If those quantum computers ever happen beyond a few qubits we may need a new encryption system.

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Yes, and it is hard to even imagine what it might be. I have been kicking around some ideas for a system that would be easily patentable. The normal trick is to multiply two huge primes p and q together to produce n, which will be part of the modulus. Pick a number e relatively prime to n as an exponent to encrypt the message, as in M^e. Then you find the inverse of e (mod φ(n)). It is this e^-1 which will be used to untangle the message on the other end. You cannot find φ(n) without knowing the factors of n. That is the immense diffculty they impose on hackers--they have to find φ(n) to get anywhere.

I am looking for a system that does not rely on factoring. Encryption is one area of math with a big, big future. I have a few wild ideas.

Maybe using larger primes will keep the current methods going until something better comes along.

I'm still putting the pieces together on the Artin conjecture. A simpler question would be "Given a number m, are there infinitely many primes p for which m is a quadratic nonresidue?" This would be a larger set since a quadratic nonresidue does not have to be a primitive root such as 8 or 12 mod 19 unless I calculated it wrong.

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Originally Posted by YesNo

Maybe using larger primes will keep the current methods going until something better comes along.

I'm still putting the pieces together on the Artin conjecture. A simpler question would be "Given a number m, are there infinitely many primes p for which m is a quadratic nonresidue?" This would be a larger set since a quadratic nonresidue does not have to be a primitive root such as 8 or 12 mod 19 unless I calculated it wrong.

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One aleph is as big as the next aleph.

I thought Aleph one was strictly bigger than Aleph null.

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Originally Posted by YesNo

I thought Aleph one was strictly bigger than Aleph null.

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Otherwise known as Aleph nought, as well, the "smallest" infinity. They are the same entity, all of them, as far as I know.

To make all possible sets from a set of size n, take 2ⁿ. The next set after aleph, is 2 raised to the aleph power. You can repeat the process over and over getting bigger sets. No one knows which of these powers has the power of the continuum.

There is an infinite heirarchy of infinities, each theoretically greater than the next, but we only have examples of two kinds. One can be thought of as the counting numbers, and the other is the uncountable points on a line, i.e. the irrataional numbers, more specifically the transcendental numbers.

The former do indeed have the cardinality of aleph, and the transcendental numbers may have the cardinality of the continuum. I believe the latter is not known for sure. Kronecker would certainly disapprove.

Impressive knowledge!

The set of functions over the reals should make a larger infinity than the reals. Which is larger than the integers or rationals. But Kronecker may be right that all of that probably doesn't matter.

You guys have learned a lot of stuff that I never did in my London school in 1956. GCE Ordinary Level. Never did math beyond that.

I wish I knew some physics and chemistry. What I know of quantum physics can be traced to someone posting something about many worlds on Lit Net and then going off to the library or the internet to try to make sense out of it. I think I know enough about many worlds at the moment to be able to reject it. The same with black holes, but I am less sure about black holes than I am about many worlds. I didn't even know about the big bang (except as some vague idea) until someone posted that the universe started from "nothing" including space and time. I got to the library as soon as I heard that. One of the most shocking moments of enlightenment was when I heard about a youtube video claiming that we never put a man on the moon. It took two days to get over that and now I'm convinced.

So now I figure if they can put man on the moon, I can solve the Artin conjecture or desiresjab can come up with a new cryptography method.

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Originally Posted by YesNo

The set of functions over the reals should make a larger infinity than the reals. Which is larger than the integers or rationals. But Kronecker may be right that all of that probably doesn't matter.

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The set of functions over the reals will not be of greater cardinality than the reals themselves. Even the transcendentals are part of the reals, and are of course as large as the whole set, in the same way that the set of even numbers is as large as all of the rationals. Cantor proved this. All that matters is whether you can map elements from one set to the other with a one-to-one correspondence. You could produce this correspondence between sets seeminlgly so sparse as the square numbers and one as dense as the rationals. They are both aleph nought in cardinality.

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Originally Posted by YesNo

I wish I knew some physics and chemistry. What I know of quantum physics can be traced to someone posting something about many worlds on Lit Net and then going off to the library or the internet to try to make sense out of it. I think I know enough about many worlds at the moment to be able to reject it. The same with black holes, but I am less sure about black holes than I am about many worlds. I didn't even know about the big bang (except as some vague idea) until someone posted that the universe started from "nothing" including space and time. I got to the library as soon as I heard that. One of the most shocking moments of enlightenment was when I heard about a youtube video claiming that we never put a man on the moon. It took two days to get over that and now I'm convinced.

So now I figure if they can put man on the moon, I can solve the Artin conjecture or desiresjab can come up with a new cryptography method.

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Good sir, none of us knows enough. The day belongs to those who seize it. Here I am at sunset trying to seize the day. Far be it from me to discourage any research. I wish you luck. Artin's conjecture is really formidable. It is one of those questions that has attracted the best quality of research. You will need to become something of an ace at modular arithmetic, since the conjecture deals with that branch, not normal algebra. The more tools you have the better you can understand what has already been done. You cannot scale Everest without some climbing gear. First, be very sure of what reciprocity means.

The guy in the link below helped me a lot with his articles. After his article on QR, the next article on biquadratic reciprocity is missing, but the ones after that are already written and posted. He ties Pytharorean triplets to reciprocity. This article gives a good idea of just how deep mere QR is, let alone general reciprocity. Hope I have not posted it before, but it is worth a read. I spent a long time on it and have probably read it ten or fifteen times. Anyone who has an easy time with this probably should have been a mathematician.

http://science.larouchepac.com/gauss...ciprocity.html

Thanks for the link. If you have any more please post them. I realize there is a lot to get familiar with before I would even know that I solved anything at all.

The first step is to show for the number 3 that there are infinitely many primes for which 3 is a quadratic nonresidue. I am sure someone has done that already. Then I would need to know what the additional conditions are to guarantee that 3 was a primitive root as well.

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Originally Posted by YesNo

Thanks for the link. If you have any more please post them. I realize there is a lot to get familiar with before I would even know that I solved anything at all.

The first step is to show for the number 3 that there are infinitely many primes for which 3 is a quadratic nonresidue. I am sure someone has done that already. Then I would need to know what the additional conditions are to guarantee that 3 was a primitive root as well.

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Yes, that could be one approach. 3 is a quadratic nonresidue of all its 4n+3 residues, and a nonresidue of all its 4n+1 nonresidues. Primes have exactly as many residues as they have nonresidues under their modulus. That seems like a decent place to start nosing around for truffles. Can you smell that truffle?

Let's see if I got this right. If I want to know if 3 is a nonresidue with respect to p, then I need to calculate (3|p)=3^1/2(p-1) mod p. If the value is -1 then it is a nonresidue. If it is 1 then it is a residue.

I might be able to get this information using quadratic reciprocity for p and q being odd primes. The formula is (p|q)(q|p) = (-1)^{1/2(p-1)*1/2(q-1)}.

Let q = 3, since that is the number I am interested in. Then 1/2(q-1) can be simplified to 1/2(3-1) = 1/2(2) = 1, so I can write the quadratic reciprocity rule as follows for q = 3.

(p|3)(3|p) = (-1)^1/2(p-1)

Now what? What I am trying to find is (3|p), but with QR I also need to find (p|3).

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Originally Posted by desiresjab

The set of functions over the reals will not be of greater cardinality than the reals themselves. Even the transcendentals are part of the reals, and are of course as large as the whole set, in the same way that the set of even numbers is as large as all of the rationals. Cantor proved this. All that matters is whether you can map elements from one set to the other with a one-to-one correspondence. You could produce this correspondence between sets seeminlgly so sparse as the square numbers and one as dense as the rationals. They are both aleph nought in cardinality.

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I'll have to think about this one and maybe do some searching.

I suspect one could assume there is a 1-1 mapping between the reals and the functions over the reals and then construct a function that is not in that set using similar substitutions that Cantor did to show that the reals are larger than the integers, however, I wonder about the details.